Degenerate facets of polytopes

Definitions and mathematical statement

A convex $d$-polytope is the convex hull of finitely many points in $\mathbb R^d$ with nonempty interior. Its vertices are its zero-dimensional faces, and a facet is a face of dimension $d-1$. A facet is degenerate in this problem when it has more than $d$ vertices; equivalently, it is not a simplex. This use of "degenerate" does not mean that the whole polytope fails to be full-dimensional.

For $n\ge d+1$ define \[ D_d(n)=\max_{P:\ f_0(P)=n} \#\{F:F\text{ is a facet of }P,\ f_0(F)>d\}. \] Determine $D_d(n)$, or sharp asymptotic bounds for fixed $d$ as $n\to\infty$. The interesting gap in the historical formulation concerns even dimensions. The source also singles out quasisimplicial polytopes, whose $(d-2)$-dimensional faces are simplices.

The subsidiary four-dimensional question asking whether $D_4(n)\ge2n$ for some $n$ has already been answered positively by the neighborly cubical constructions reported in the source's update. It is therefore not posed here as a fresh open conjecture. The definition records the remaining extremal-count question, including its quasisimplicial restriction.

Short English statement

How many nonsimplicial facets can a convex polytope with a prescribed number of vertices have? The old threshold of twice as many facets as vertices in dimension four is already surpassed.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3800003 (AMR-037-0003), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Erickson - Open Problems in Computational Geometry, Degenerate facets of polytopes. Original source indexed by AMR-037-0003. https://jeffe.cs.illinois.edu/open/.
• S3 Jeff Erickson, Open Combinatorial Problems, degenerate facets and the 1999 update. https://jeffe.cs.illinois.edu/open/comb.html.